Triplet down conversion, the process of converting one high-energy photon into three low-energy photons, may soon be experimentally feasible in the optical regime, due to advances in optical resonator technology. We use quantum phase-space techniques to analyze the process of degenerate intracavity triplet down conversion by solving stochastic differential equations within the truncated positive-P representation. We simulate the time evolution of both intracavity fields, and examine the resulting steady-states as a function of the pump intensity. Quantum effects are most pronounced in the region immediately above the semi-classical pumping threshold, where our numerical results differ significantly from semi-classical predictions. We calculate steady-state fluctuation spectra and identify regimes of squeezing, bipartite entanglement, and non-Gaussianity measurable in the cavity output fields. We also validate the truncated positive-P description against Monte Carlo wave function simulations, finding good agreement for low mode populations.